Geology Reference
In-Depth Information
Now consider the prediction error power of the ( N
+
1)-length prediction error
filter. It is
1 g k f j k
E
N
N
E j j
f j
1 g l f j l
P N + 1
=
=
f j
k
=
l
=
= E
k = 1 g k f j k
l = 1 g l f j l f j
N
N
f j f j f j
E
N
N
1 g l f j l
+
1 g k f j k
k
=
l
=
l = 1 g l E f j f j l
N
k = 1 g k E f j f j k
N
= φ ff ( 0 )
N
N
1 g k g l E f j k f j l .
+
(2.76)
k =
1
l =
Thus,
l = 1 g l φ ff ( l )
k = 1 g k φ ff ( k )
N
N
P N + 1 = φ ff (0)
N
N
1 g k g l φ ff ( l
+
k ).
(2.77)
k
=
1
l
=
From the unit prediction equations (2.73), the last term on the right-hand side of
expression (2.77) can be transformed to give
l = 1 g l φ ff ( l )
k = 1 g k φ ff ( k )
l = 1 g l φ ff ( l )
N
N
N
= φ ff ( 0 )
P N + 1
+
k = 1 g k φ ff ( k )
N
k = 0 γ k φ ff ( k )
N
k = 0 γ k φ ff (
N
= φ ff (0)
=
=
k ).
(2.78)
Writing the unit prediction equations (2.73) in terms of the prediction error coef-
ficients they become
N
0 γ k φ ff ( m
k )
=
0, m
=
1,..., N .
(2.79)
k
=
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